An Algebra Problem from RMO

Algebra Level 3

\[\begin{cases} f(x)={ x }^{ 3 }+{ a }x^{ 2 }+bx+c \\ g(x)={ x }^{ 3 }+{ b }x^{ 2 }+cx+a \end{cases}\] Given that \(a, b, c\) are integers with \(c\neq 0\) and the following conditions hold for the above functions:

(a) \(f(1) = 0\);

(b) the roots of \(g(x) = 0\) are the squares of the roots of \(f(x) = 0\).

Find the absolute value of \[({ a }^{ 2013 }+{ b }^{ 2013 }+{ c }^{ 2013 }).\]

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